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Properties

Label 2-72-72.13-c1-0-9
Degree $2$
Conductor $72$
Sign $-0.130 + 0.991i$
Analytic cond. $0.574922$
Root an. cond. $0.758236$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.179 − 1.40i)2-s + (0.986 − 1.42i)3-s + (−1.93 + 0.504i)4-s + (−1.19 − 0.687i)5-s + (−2.17 − 1.12i)6-s + (1.80 + 3.12i)7-s + (1.05 + 2.62i)8-s + (−1.05 − 2.80i)9-s + (−0.750 + 1.79i)10-s + (1.83 − 1.05i)11-s + (−1.19 + 3.25i)12-s + (−0.887 − 0.512i)13-s + (4.06 − 3.09i)14-s + (−2.15 + 1.01i)15-s + (3.49 − 1.95i)16-s + 0.808·17-s + ⋯
L(s)  = 1  + (−0.127 − 0.991i)2-s + (0.569 − 0.822i)3-s + (−0.967 + 0.252i)4-s + (−0.532 − 0.307i)5-s + (−0.887 − 0.460i)6-s + (0.682 + 1.18i)7-s + (0.373 + 0.927i)8-s + (−0.351 − 0.936i)9-s + (−0.237 + 0.567i)10-s + (0.552 − 0.319i)11-s + (−0.343 + 0.939i)12-s + (−0.246 − 0.142i)13-s + (1.08 − 0.826i)14-s + (−0.556 + 0.262i)15-s + (0.872 − 0.487i)16-s + 0.196·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.130 + 0.991i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ -0.130 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.603924 - 0.688632i\)
\(L(\frac12)\) \(\approx\) \(0.603924 - 0.688632i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.179 + 1.40i)T \)
3 \( 1 + (-0.986 + 1.42i)T \)
good5 \( 1 + (1.19 + 0.687i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.80 - 3.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.83 + 1.05i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.887 + 0.512i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.808T + 17T^{2} \)
19 \( 1 - 7.43iT - 19T^{2} \)
23 \( 1 + (1.65 - 2.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.71 - 4.45i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.26 + 5.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.01iT - 37T^{2} \)
41 \( 1 + (3.45 - 5.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.245 - 0.142i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.61 + 6.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.86iT - 53T^{2} \)
59 \( 1 + (-7.06 - 4.08i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.31 + 3.64i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.43 + 1.40i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.69T + 71T^{2} \)
73 \( 1 - 0.409T + 73T^{2} \)
79 \( 1 + (0.0456 + 0.0790i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.40 - 1.39i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.91T + 89T^{2} \)
97 \( 1 + (2.76 + 4.78i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.25882852741799366713920528241, −12.96361828520117407246811989733, −12.01962611044808486056117478712, −11.55877124551154408946280671403, −9.687675386685760306925843625451, −8.539703991784501036174565693407, −7.88387243578902462271828246565, −5.65338999169909647334177879957, −3.68930295527200324136076743947, −1.90477263585986738383773588277, 3.90102024084730398680336707285, 4.87546041751562841196239980775, 6.98719425789938721107315552472, 7.88492660428633097179564135335, 9.107512481571810614785397136960, 10.24334137642268194022638861089, 11.35090225813189974519359573098, 13.35274119146532110058307530158, 14.21737514296421696754140492238, 14.98171010821158257131000638943

Graph of the $Z$-function along the critical line